Determine the general solution of
Sin2x=4cos2x
thanks in advance
sin 2x/cos 2x = 4
tan 2x = 4
since tan is +, 2x is in either quads I or III
2x = 75.964° or 2x = 180+75.964 = 255.964°
x = 37.98° or x = 127.98
since the period of tan 2x is 90°
general solution:
x = 37.98 + 90k
x = 127.98 + 90k
since when k=1 we hit 127.98
we can use only x = 37.98 + 90k
I will leave it up to you to set your calculator to radians, if the answer is needed in those units,
you would then add (π/2)k
I love the way the question was answered thanks for the explanation
To find the general solution of the equation sin(2x) = 4cos(2x), we'll need to use trigonometric identities and solve for all possible values of x.
First, we will convert sin(2x) to its equivalent expression using the double-angle identity:
sin(2x) = 2sin(x)cos(x)
Now, the equation becomes:
2sin(x)cos(x) = 4cos(2x)
We can simplify this equation by dividing both sides by 2cos(x):
sin(x) = 2cos(2x)/cos(x)
Next, we will use the identity: cos(2x) = 1 - 2sin^2(x) to substitute for cos(2x):
sin(x) = 2(1 - 2sin^2(x))/cos(x)
Multiplying both sides by cos(x), we get:
sin(x)cos(x) = 2(1 - 2sin^2(x))
Using the identity: sin(2x) = 2sin(x)cos(x), the equation becomes:
sin(2x) = 2 - 4sin^2(x)
Rearranging the terms, we have:
4sin^2(x) + sin(2x) - 2 = 0
Now, let's use a substitution to simplify the equation. Let u = sin(x), then we have:
4u^2 + sin(2x) - 2 = 0
Using the double-angle identity, sin(2x) = 2sin(x)cos(x), we can rewrite the equation as:
4u^2 + 2u - 2 = 0
Next, we can solve this quadratic equation by factoring or using the quadratic formula. In this case, we'll use the quadratic formula:
u = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the values into the formula, we have:
u = (-(2) ± √((2)^2 - 4(4)(-2))) / (2(4))
u = (-2 ± √(4 + 32)) / 8
u = (-2 ± √36) / 8
Simplifying further, we get:
u = (-2 ± 6) / 8
u = 1/2 or -1/2
Since u = sin(x), we can solve for x:
sin(x) = 1/2 or sin(x) = -1/2
For sin(x) = 1/2, the solutions are x = π/6 + 2πn and x = 5π/6 + 2πn, where n is an integer.
For sin(x) = -1/2, the solutions are x = 7π/6 + 2πn and x = 11π/6 + 2πn, where n is an integer.
Therefore, the general solution for the equation sin(2x) = 4cos(2x) is:
x = π/6 + 2πn, x = 5π/6 + 2πn, x = 7π/6 + 2πn, and x = 11π/6 + 2πn, where n is an integer.
To determine the general solution of the equation sin(2x) = 4cos(2x), we'll use the trigonometric identity that relates sin(2x) and cos(2x).
The identity states that sin(2x) = 2 sin(x) cos(x) and cos(2x) = cos^2(x) - sin^2(x).
So, we can substitute these identities into the given equation:
2sin(x)cos(x) = 4(cos^2(x) - sin^2(x))
Expanding the equation, we get:
2sin(x)cos(x) = 4cos^2(x) - 4sin^2(x)
Rearranging the equation, we have:
4sin^2(x) + 2sin(x)cos(x) - 4cos^2(x) = 0
Now, we can factor the equation:
(2sin(x) + 4cos(x))(2sin(x) - cos(x)) = 0
Setting each factor to zero, we have two separate equations:
1) 2sin(x) + 4cos(x) = 0
2) 2sin(x) - cos(x) = 0
Now we solve each equation separately:
1) 2sin(x) + 4cos(x) = 0
Dividing both sides by 2, we get:
sin(x) + 2cos(x) = 0
Rearranging, we have:
sin(x) = -2cos(x)
Taking the ratio of both sides, we have:
tan(x) = -2
Using a calculator or a trigonometric table, we find the principal solution angle to be approximately -63.43 degrees or -1.11 radians.
2) 2sin(x) - cos(x) = 0
Dividing both sides by 2, we get:
sin(x) - (1/2)cos(x) = 0
Rearranging, we have:
sin(x) = (1/2)cos(x)
Taking the ratio of both sides, we have:
tan(x) = 1/2
Using a calculator or a trigonometric table, we find the principal solution angle to be approximately 26.57 degrees or 0.47 radians.
Therefore, the general solution for the equation sin(2x) = 4cos(2x) is:
x = -1.11 + n(180 degrees) or x = 0.47 + n(180 degrees)
where n is an integer representing all possible solutions.